IDEALS AND CONTINUITY FOR QUANTALOID-ENRICHED CATEGORIES.

We study ideals in, and continuity of, quantaloid-enriched categories (Qcategories for short) as a 'many-valued and many-typed' generalization of domain theory. Abstractly, for any (saturated) class Φ of presheaves, we define and study the Φ-continuity of Q-categories. Concretely, we compute three examples of such saturated classes of presheaves - the class of flat ideals, the class of irreducible ideals and the class of conical ideals - which are proper generalizations of ideals in domain theory. [ABSTRACT FROM AUTHOR]